Several Amazing Discoveries About Compact Metrizable Spaces in ZF

arXiv:2008.01233v1 [math.GN] 3 Aug 2020 eateto ttsisadAtailFnnilMathema Actuarial-Financial and Statistics of Department eea mzn icvre bu compact about discoveries amazing Several nvriyo h een alvsi820 ao,Greece Samos, 83200, Karlovassi Aegean, the of University yiksKrmds lfhro ahssadEiaWajch Eliza and Tachtsis Eleftherios Keremedis, Kyriakos ae.Sm ftesaeet r rvbein provable are statements is the spaces of compact Some metrizable gated. about statements natural many eidpnetof independent be aeacoc ucin n opc erzbesaene n need fa space can metrizable cube compact sets Tychonoff the a finite into and non-empty embeddable function, of choice family a denumerable have a of each but in isfied constructed of are set models power the symmetric New applied. are ZF eateto ahmtc,Uiest fteAegean the of University Mathematics, of Department idc nvriyo aua cecsadHumanities and Sciences Natural of University Siedlce nteasneo h xo fcoc,testtertcstat set-theoretic the choice, of axiom the of absence the In n emtto oesof models permutation and aut fEatadNtrlSciences Natural and Exact of Faculty l aa5,0-1 idc,Poland Siedlce, 08-110 54, Maja 3 ul. erzbesae in spaces metrizable alvsi ao 30,Greece 83200, Samos Karlovassi, R swl-real,teCniumHptei ssat- is Hypothesis Continuum the well-orderable, is ZF nttt fMathematics of Institute o neednersls itntmdl of models distinct results, independence For . [email protected] [email protected] uut5 2020 5, August [email protected] Abstract ZFA 1 ihtase hoeso Pincus of theorems transfer with [0 , 1] R . ZF oeaesonto shown are some , ZF investi- which tbe ot sof us lto il tics, Mathematics Subject Classification (2010): 03E25, 03E35, 54A35, 54E35, 54D30 Keywords: Weak forms of the Axiom of Choice, metrizable space, totally bounded metric, compact space, permutation model, symmetric model.

1 Preliminaries

1.1 The set-theoretic framework

In this paper, the intended context for reasoning and statements of theorems is the Zermelo-Fraenkel set theory ZF without the axiom of choice AC. The system ZF + AC is denoted by ZFC. We recommend [32] and [33] as a good introduction to ZF. To stress the fact that a result is proved in ZF or ZF + A (where A is a statement independent of ZF), we shall write at the beginning of the statements of the theorems and propositions (ZF) or (ZF + A), respectively. Apart from models of ZF, we refer to some models of ZFA (or ZF0 in [15]), that is, we refer also to ZF with an infinite set of atoms (see [20], [21] and [15]). Our theorems proved here in ZF are also provable in ZFA; however, we also mention some theorems of ZF that are not theorems of ZFA. We denote by ω the set of all non-negative integers (i.e., finite ordinal numbers of von Neumann). As usual, if n ∈ ω, then n +1 = n ∪ {n}. Members of the set N = ω \{0} are called natural numbers. The power set of a set X is denoted by P(X). A set X is called countable if X is equipotent to a subset of ω. A set X is called uncountable if X is not countable. A set X is finite if X is equipotent to an element of ω. An infinite set is a set which is not finite. An infinite countable set is called denumerable. A cardinal number of von Neumann is an initial ordinal number of von Neumann. If X is a set and κ is a non-zero cardinal number of von Neumann, then [X]κ is the family of all subsets of X equipotent to κ, [X]≤κ is the collection of all subsets of X equipotent to subsets of κ, and [X]<κ is the family of all subsets of X equipotent to a cardinal number of von Neumann which belongs to κ. For a set X, we denote by |X| the cardinal number of X in the sense of Definition 11.2 of [20]. We recall that, in ZF, for every set X, the cardinal number |X| exists; however, X is equipotent to a cardinal number of von Neumann if and only if X is well-orderable. For sets X and Y , the inequality |X|≤|Y |

2 means that X is equipotent to a subset of Y . The set of all real numbers is denoted by R and, if it is not stated otherwise, R and every subspace of R are considered with the usual topology and with the metric induced by the standard absolute value on R.

1.2 Notation and basic deﬁnitions In this subsection, we establish notation and recall several basic deﬁnitions. Let X = hX,di be a metric space. The d-ball with centre x ∈ X and radius r ∈ (0, +∞) is the set

Bd(x, r)= {y ∈ X : d(x, y)

The collection 1 τ(d)= {V ⊆ X :(∀x ∈ V )(∃n ∈ ω)B (x, ) ⊆ V } d 2n is the topology in X induced by d. For a set A ⊆ X, let δd(A)=0 if A = ∅, and let δd(A) = sup{d(x, y): x, y ∈ A} if A =6 ∅. Then δd(A) is the diameter of A in X.

Deﬁnition 1.1. Let X = hX,di be a metric space.

(i) Given a real number ε > 0, a subset D of X is called ε-dense or an ε-net in X if X = Bd(x, ε). xS∈D (ii) X is called totally bounded if, for every real number ε> 0, there exists a ﬁnite ε-net in X.

(iii) X is called strongly totally bounded if it admits a sequence (Dn)n∈N N 1 X such that, for every n ∈ , Dn is a ﬁnite n -net in . (iv) (Cf. [24].) d is called strongly totally bounded if X is strongly totally bounded.

Remark 1.2. Every strongly totally bounded metric space is evidently totally bounded. However, it was shown in [24, Proposition 8] that the sentence “Every totally bounded metric space is strongly totally bounded” is not a theorem of ZF.

3 Deﬁnition 1.3. Let X = hX, τi be a topological space and let Y ⊆ X. Suppose that B is a base of X.

(i) The closure of Y in X is denoted by clτ (Y ) or clX(Y ).

(ii) τ|Y = {U ∩ Y : U ∈ τ}. Y = hY, τ|Y i is the subspace of X with the underlying set Y .

(iii) BY = {U ∩ Y : U ∈ B}.

Clearly, in Deﬁnition 1.3 (iii), BY is a base of Y. In Section 5, it is shown that BY need not be equipotent to a subset of B. In the sequel, boldface letters will denote metric or topological spaces (called spaces in abbreviation) and lightface letters will denote their underlying sets.

Deﬁnition 1.4. A collection U of subsets of a space X is called:

(i) locally ﬁnite if every point of X has a neighbourhood meeting only ﬁnitely many members of U;

(ii) point-ﬁnite if every point of X belongs to at most ﬁnitely many members of U;

(iii) σ-locally finite (respectively, σ-point-finite) if U is a countable union of locally finite (respectively, point-finite) subfamilies.

Deﬁnition 1.5. A space X is called:

(i) ﬁrst-countable if every point of X has a countable base of neighbour- hoods;

(ii) second-countable if X has a countable base.

Given a collection {Xj : j ∈ J} of sets, for every i ∈ J, we denote by πi the projection πi : Xj → Xi defined by πi(x) = x(i) for each x ∈ Xj. jQ∈J jQ∈J If τj is a topology in Xj, then X = Xj denotes the Tychonoff product of jQ∈J the topological spaces Xj = hXj, τji with j ∈ J. If Xj = X for every j ∈ J, J then X = Xj. As in [8], for an infinite set J and the unit interval [0, 1] jQ∈J of R, the cube [0, 1]J is called the Tychonoff cube. If J is denumerable, then

4 the Tychonoff cube [0, 1]J is called the Hilbert cube. In [12], all Tychonoff cubes are called Hilbert cubes. In [42], Tychonoff cubes are called cubes. We recall that if Xj =6 ∅, then it is said that the family {Xj : j ∈ J} jQ∈J has a choice function, and every element of Xj is called a choice function jQ∈J of the family {Xj : j ∈ J}. A multiple choice function of {Xj : j ∈ J} is every function f ∈ P(Xj) such that, for every j ∈ J, f(j) is a non- jQ∈J empty finite subset of Xj. A set f is called partial (multiple) choice function of {Xj : j ∈ J} if there exists an infinite subset I of J such that f is a (multiple) choice function of {Xj : j ∈ I}. Given a non-indexed family A, we treat A as an indexed family A = {x : x ∈ A} to speak about a (partial) choice function and a (partial) multiple choice function of A. Let {Xj : j ∈ J} be a disjoint family of sets, that is, Xi ∩ Xj = ∅ for each pair i, j of distinct elements of J. If τj is a topology in Xj for every j ∈ J, then Xj denotes the direct sum of the spaces Xj = hXj, τji with j ∈ J. jL∈J

Deﬁnition 1.6. (Cf. [2], [34] and [26].)

(i) A space X is said to be Loeb (respectively, weakly Loeb) if the family of all non-empty closed subsets of X has a choice function (respectively, a multiple choice function).

(ii) If X is a (weakly) Loeb space, then every (multiple) choice function of the family of all non-empty closed subsets of X is called a (weak) Loeb function of X.

Other topological notions used in this article but not deﬁned here are standard. They can be found, for instance, in [8] and [42].

Deﬁnition 1.7. A set X is called:

(i) a cuf set if X is expressible as a countable union of ﬁnite sets (cf. [5], [6], [19] and [16, Form 419]);

(ii) Dedekind-finite if X is not equipotent to a proper subset of itself (cf. [15, Note 94], [12, Definition 4.1] and [20, Definition 2.6]); Dedekind- infinite if X is not Dedekind-finite (equivalently, if there exists an injection f : ω → X) (cf. [15, Note 94] and [12, Definition 2.13]);

5 (iii) amorphous if X is infinite and there does not exist a partition of X into two infinite sets (cf. [15, Note 57],[20, p. 52] and [12, E. 11 in Section 4.1]). Definition 1.8. (Cf. [31].) A topological space hX, τi is called a cuf space if X is a cuf set.

1.3 The list of weaker forms of AC In this subsection, for readers’ convenience, we define and denote most of the weaker forms of AC used directly in this paper. If a form is not defined in the forthcoming sections, its definition can be found in this subsection. For the known forms given in [15], [16] or [12], we quote in their statements the form number under which they are recorded in [15] (or in [16] if they do not appear in [15]) and, if possible, we refer to their definitions in [12].

Deﬁnition 1.9. 1. ACfin ([15, Form 62]): Every non-empty family of non-empty ﬁnite sets has a choice function.

2. ACWO ( [15, Form 60]): Every non-empty family of non-empty well- orderable sets has a choice function.

3. CAC ([15, Form 8], [12, Deﬁnition 2.5]): Every denumerable family of non-empty sets has a choice function.

4. CAC(R) ([15, Form 94], [12, Deﬁnition 2.9(1)]): Every denumerable family of non-empty subsets of R has a choice function.

5. CAC△ω(R) (Cf. [29]): For every family A = {An : n ∈ ω} such that, <ω for every n ∈ ω and all x, y ∈ An, ∅= 6 An ⊆ P(ω)\{∅} and x△y ∈ [ω] (△ denotes the operation of symmetric diﬀerence between sets), there exists a choice function of A.

6. IDI ([15, Form 9], [12, Definition 2.13(ii)]): Every Dedekind-finite set is finite.

7. IDI(R) ([15, Form 13], [12, Definition 2.13(2)]): Every Dedekind-finite subset of R is finite.

8. WoAm ([15, Form 133]): Every set is either well-orderable or has an amorphous subset.

6 9. Part(R): Every partition of R is of size ≤|R|.

10. WO(R) ([15, Form 79]): R is well-orderable.

11. WO(P(R)) ([15, Form 130]): P(R) is well-orderable.

12. CACfin ([15, Form 10], [12, Deﬁnition 2.9(3)]): Every denumerable family of non-empty ﬁnite sets has a choice function.

13. For a ﬁxed n ∈ ω \{0, 1}, CACn ([15, Form 288(n)]): Every denumerable family of n-element sets has a choice function.

14. CACWO: Every denumerable family of non-empty well-orderable sets has a choice function.

15. CMC ([15, Form 126], [12, Deﬁnition 2.10]): Every denumerable family of non-empty sets has a multiple choice function.

16. CMCω ([15, Form 350]): Every denumerable family of denumerable sets has a multiple choice function.

17. CUC ([15, Form 31], [12, Deﬁnition 3.2(1)]): Every countable union of countable sets is countable.

18. CUCfin ([15, Form [10 A]], [12, Deﬁnition 3.2(3)]): Every countable union of ﬁnite sets is countable.

19. UT(ℵ0,cuf,cuf) ([16, Form 419]): Every countable union of cuf sets is a cuf set. (Cf. also [6].)

20. UT(ℵ0, ℵ0,cuf) ([16, Form 420]): Every countable union of countable sets is a cuf set. (Cf. also [6].)

21. vDCP(ℵ0) ( [15, Form 119], [12, p, 79], [7]): Every denumerable family {hAn, ≤ni : n ∈ ω} of linearly ordered sets, each of which is order- isomorphic to the set hZ, ≤i of integers with the standard linear order ≤, has a choice function.

22. BPI ([15, Form 14], [12, Deﬁnition 2.15(1)]): Every Boolean algebra has a prime ideal.

7 23. DC ([15, Form 43], [12, Deﬁnition 2.11(1)]): For every non-empty set X and every binary relation ρ on X if, for each x ∈ X there exists y ∈ X such that xρy, then there exists a sequence (xn)n∈N of points of X such that xnρxn+1 for each n ∈ N. Remark 1.10. The following are well-known facts in ZF:

(i) CACfin and CUCfin are both equivalent to the sentence: Every infinite well-ordered family of non-empty finite sets has a partial choice function (see [15, Form [10 O]] and [12, Diagram 3.4, p. 23]). Moreover, CACfin is equivalent to Form [10 E] of [15], that is, to the sentence: Every denumerable family of non-empty finite sets has a partial choice function. It is known that IDI implies CACfin and this implication is not reversible in ZF (cf. [12, pp. 324–324]). (ii) CAC is equivalent to the sentence: Every denumerable family of non- empty sets has a partial choice function (see [15, Form [8 A]]). (iii) BPI is equivalent to the statement that all products of compact Haus- dorff spaces are compact (see [15, Form [14 J]] and [12, Theorem 4.37]).

(iv) CMCω is equivalent to the following sentence: Every denumerable family of denumerable sets has a multiple choice function. Remark 1.11. (a) It was proved in [19] that the following implications are true in ZF and none of the implications is reversible in ZF:

CMC → UT(ℵ0,cuf,cuf) → CMCω → vDCP(ℵ0).

(b) Clearly, UT(ℵ0,cuf,cuf) implies UT(ℵ0, ℵ0,cuf). In [6, proof to The- orem 3.3] a model of ZFA was shown in which UT(ℵ0, ℵ0,cuf) is true and UT(ℵ0,cuf,cuf) is false. (c) It was proved in [31] that the following equivalences hold in ZF:

(i) UT(ℵ0,cuf,cuf) is equivalent to the sentence: Every countable product of one-point Hausdorff compactifications of infinite discrete cuf spaces is metrizable (equivalently, first-countable).

(ii) UT(ℵ0, ℵ0,cuf) is equivalent to the sentence: Every countable product of one-point Hausdorff compactifications of denumerable discrete spaces is metrizable (equivalently, first-countable).

8 Let us pass to deﬁnitions of forms concerning metric and metrizable spaces.

Deﬁnition 1.12. 1. CAC(R,C): For every disjoint family A = {An : n ∈ N} of non-empty subsets of R, if there exists a family {dn : n ∈ N} of metrics such that, for every n ∈ N, hAn,dni is a compact metric space, then A has a choice function.

2. CAC(C, M): If {hXn,dni : n ∈ ω} is a family of non-empty compact metric spaces, then the family {Xn : n ∈ ω} has a choice function. 3. M(TB,WO): For every totally bounded metric space hX,di, the set X is well-orderable.

4. M(TB,S): Every totally bounded metric space is separable.

5. ICMDI: Every inﬁnite compact metrizable space is Dedekind-inﬁnite.

6. MP ([15, Form 383]): Every metrizable space is paracompact.

7. M(σ − p.f.) ([15, Form 233]): Every metrizable space has a σ-point- ﬁnite base.

8. M(σ−l.f.) ([15, Form [232 B]]): Every metrizable space has a σ-locally ﬁnite base.

Deﬁnition 1.13. The following forms will be called forms of type M(C, ).

1. M(C,S): Every compact metrizable space is separable.

2. M(C, 2): Every compact metrizable space is second-countable.

3. M(C,STB): Every compact metric space is strongly totally bounded.

4. M(C, L): Every compact metrizable space is Loeb.

5. M(C,WO): Every compact metrizable space is well-orderable.

M(C, ֒→ [0, 1]N): Every compact metrizable space is embeddable in the .6 Hilbert cube [0, 1]N.

M(C, ֒→ [0, 1]R): Every compact metrizable space is embeddable in the .7 Tychonoﬀ cube [0, 1]R.

9 8. M(C, ≤|R|): Every compact metrizable space is of size ≤|R|.

9. M(C, W (R)): For every inﬁnite compact metrizable space hX, τi, τ and R are equipotent.

10. M(C, B(R)): Every compact metrizable space has a base of size ≤|R|.

11. M(C, |BY | ≤ |B|): For every compact metrizable space X, every base B of X and every compact subspace Y of X, |BY | ≤ |B|.

12. M([0, 1], |BY | ≤ |B|): For every base B of the interval [0, 1] with the usual topology and every compact subspace Y of [0, 1], |BY | ≤ |B|.

13. M(C, σ − l.f): Every compact metrizable space has a σ-locally ﬁnite base.

14. M(C, σ − p.f): Every compact metrizable space has a σ-point-ﬁnite base.

The notation of type M(C, ) was started in [22] and continued in [23] but not all forms from the definition above were defined in [22] and [23]. The forms M(C, L) and M(C,WO) were denoted by CML and CMWO in [27]. →֒ ,Most forms from Definition 1.13 are new here. That the new forms M(C N R M(C, ֒→ [0, 1] ), M(C, W (R)), M(C, B(R)), M(C, |BY | ≤ |B|) and ,( [1 ,0] M([0, 1], |BY | ≤ |B|) are all important is shown in Section 4. Apart from the forms defined above, we also refer to the following forms that are not weaker than AC in ZF:

Deﬁnition 1.14. 1. LW ([15, Form 90]): For every linearly ordered set hX, ≤i, the set X is well-orderable.

ℵ0 2. CH (the Continuum Hypothesis): 2 = ℵ1.

Remark 1.15. It is well known that AC and LW are equivalent in ZF; however, LW does not imply AC in ZFA (see [15] and [20, Theorems 9.1 and 9.2]).

10 2 Introduction

2.1 The content of the article in brief

Although mathematicians are aware that a lot of theorems of ZFC that are included in standard textbooks on general topology (e.g., in [8] and [42]) may fail in ZF and many amazing disasters in topology in ZF have been dis- covered, new non-trivial results showing significant differences between truth values in ZFC and in ZF of some given propositions can be still surprising. In this article, we show new results concerning forms of type M(C, ) in ZF. The main aim of our work is to establish in ZF the set-theoretic strength of the forms of type M(C, ), as well as relationships between these forms and relevant ones. Taking care of the readability of the article, in the forthcoming Subsections 2.2–2.4, we include some known facts and few definitions for future references. In particular, in Subsection 2.4, we give definitions of permutation models (called also Fraenkel-Mostowski models) and formulate only this version of a transfer theorem due to Pincus (called here the Pincus Transfer Theorem) (cf. Theorem 2.19) which is applied in this article. The main new results of the article are included in Sections 3–5. Section 6 con- tains a list of open problems that suggest a direction for future research in this field. In Section 3, we construct new symmetric ZF-models in each of which the conjunction CH ∧ WP(P(R)) ∧ ¬CACfin is true. In Section 4, we investigate relationships between the forms M(TB,WO), M(TB,S), ICMDI and M(C,S). Among other results of Section 4, by showing appropriate permutation models and using the Pincus Transfer The- orem, we prove that the conjunctions BPI ∧ ICMDI ∧ ¬IDI, (¬BPI) ∧ ICMDI ∧ ¬IDI and UT(ℵ0, ℵ0,cuf) ∧ ¬ICMDI have ZF-models (see The- orems 4.13, 4.14 and 4.23, respectively). We deduce that the conjunction UT(ℵ0, ℵ0,cuf) ∧ ¬M(C,S) has a ZF-model (see Corollary 4.24). We dis- cuss a relationship between M(C,S) and CUC. Taking the opportunity, we fill in a gap in [15] and [16] by proving that WoAm implies CUC (see Proposition 4.18). Among a plethora of results of Section 5, we show that CACfin implies neither M(C,S) nor M(C, ≤ |R|) (see Proposition 5.1), and M(C,S) is equivalent to each one of the conjunctions: CACfin ∧ M(C, σ − l.f.), CACfin ∧ M(C,STB) and CAC(R,C) ∧ M(C, ≤ |R|) (see Theorems 5.2 and 5.4, respectively). We deduce that M(C, σ − l.f) is unprovable in ZF

11 (see Remark 5.3). We also deduce that M(C,S) and M(C, ≤|R|) are equivalent in every permutation model (see Corollary 5.5). Furthermore, we prove that M(C,S) and M(C, ֒→ [0, 1]N) are equivalent (see Theorem 5.8). We show that, surprisingly, M(C, |BY | ≤ |B|) and M(C, B(R)) are independent of ZF (see Theorem 5.11). In Theorem 5.12, we show that M(C, B(R)) is equivalent to the conjunction M(C, ֒→ [0, 1]R) ∧ Part(R), CAC(R) implies that M(C,S) and M(C, B(R)) are equivalent; moreover, the statement that M(C,S) and M(C, W (R)) are equivalent is equivalent to M(C, B(R)). The models of ZF + WO(P(R))+ ¬CACfin constructed in Section 3 are applied to a proof that Part(R) does not imply M(C, ֒→ [0, 1]R) in ZF (see Theorem 5.13).

2.2 A list of several known theorems We list below some known theorems for future references.

Theorem 2.1. (Cf. [36].) CAC implies M(TB,S) in ZF.

Theorem 2.2. (Cf. [23].)(ZF)

(i) Let X = hX,di be an uncountable compact separable metric space. Then |X| = |R|.

(ii) CACfin follows from each of the statements: M(C,S), M(C, ≤ |R|) and “For every compact metric space hX,di, either |X|≤|R| or |R| ≤ |X|”.

Theorem 2.3. (ZF)

(a) ([28, Theorem 8].) The statements M(C,S), CAC(C, M) are equivalent.

(b) ([28, Corollary 1(a)].) CAC(C, M) implies CACfin.

Theorem 2.4. ([9, Corollary 4.8], Urysohn’s Metrization Theorem.) (ZF) If X is a second-countable T3-space, then X is metrizable.

Theorem 2.5. (Cf. [35], [39], [1], [3]. )

(i) (ZFC) Every metrizable space has a σ-locally ﬁnite base.

12 (ii) (ZF) If a T1-space X is regular and has a σ-locally ﬁnite base, then X is metrizable.

Remark 2.6. That it holds in ZFC that a T1-space is metrizable if and only if it is regular and has a σ-locally finite base was originally proved by Nagata in [35], Smirnov in [39] and Bing in [1]. It was shown in [3] that it is provable in ZF that every regular T1-space which admits a σ-locally finite base is metrizable. It was established in [14] that M(σ − l.f.) is an equivalent to M(σ − p.f.) and implies MP. Using similar arguments, one can prove that M(C, σ − l.f.) and M(C, σ − p.f) are also equivalent in ZF. In [10], a model of ZF + DC was shown in which MP fails. In [4], a model of ZF + BPI was shown in which MP fails. This implies that, in each of the above- mentioned ZF- models constructed in [10] and [4], there exists a metrizable space which fails to have a σ-point-finite base. This means that M(σ − l.f) is unprovable in ZF. In Section 4, it is clearly explained that M(C, σ − f.l) is also unprovable in ZF.

Theorem 2.7. (ZF)

(i) (Cf. [27].) A compact metrizable space is Loeb iﬀ it is second-countable iﬀ it is separable. In consequence, the statement M(C, L), M(C,S) and M(C, 2) are all equivalent.

(ii) (Cf. [30].) If X is a compact second-countable and metrizable space, then Xω is compact and separable. In particular, the Hilbert cube [0, 1]N is a compact, separable metrizable space.

(iii) (Cf. [23].) BPI implies M(C,S) and M(C, ≤|R|).

2.3 Frequently used metrics

Similarly to [31], we make use of the following idea several times in the sequel. Suppose that A = {An : n ∈ N} is a disjoint family of non-empty sets, A = An and ∞ ∈/ A. Let X = A ∪ {∞}. Suppose that (ρn)n∈N is a nS∈N sequence such that, for each n ∈ N, ρn is a metric on An. Let dn(x, y) = 1 R min{ρn(x, y), n } for all x, y ∈ An. We deﬁne a function d : X × X → as follows:

13 0 if x = y;  max{ 1 , 1 } if x ∈ A ,y ∈ A and n =6 m;  n m n m (∗) d(x, y)=  dn(x, y) if x, y ∈ An; 1  n if x ∈ A and y = ∞ or x = ∞ and y ∈ A.  Proposition 2.8. The function d, deﬁned by (∗), has the following properties:

(i) d is a metric on X (cf. [31]);

(ii) if, for every n ∈ N, the space hAn, τ(ρn)i is compact, then so is the space hX, τ(d)i (cf. [31]);

(iii) the space hX, τ(d)i has a σ-locally ﬁnite base;

(iv) if A does not have a choice function, the space hX, τ(d)i is not separable.

Metrics deﬁned by (∗) were used, for instance, in [26], [27], [31], as well as in several other papers not cited here.

2.4 Permutation models and the Pincus Transfer The- orem Let us clarify deﬁnitions of the permutation models we deal with. We refer to [20, Chapter 4] and [21, Chapter 15, p. 251] for the basic terminology and facts concerning permutation models. Suppose we are given a model M of ZFA + AC with an inﬁnite set A of all atoms of M, and a group G of permutations of A. For a set x ∈M, we denote by TC(x) the transitive closure of x in M. Then every permutation φ of A extends uniquely to an ∈-automorphism (usually denoted also by φ) of M. For x ∈M, we put:

ﬁxG(x)= {φ ∈ G :(∀t ∈ x)φ(t)= t} and symG(x)= {φ ∈ G : φ(x)= x}.

We refer the readers to [20, Chapter 4, pp. 46–47] for the deﬁnitions of the concepts of a normal ﬁlter and a normal ideal.

14 Definition 2.9. (i) The permutation model N determined by M, G and a normal filter F of subgroups of G is defined by the equality:

N = {x ∈M :(∀t ∈ TC({x}))(symG(t) ∈F)}.

(ii) The permutation model N determined by M, G and a normal ideal I of subsets of the set of all atoms of M is deﬁned by the equality:

N = {x ∈M :(∀t ∈ TC({x}))(∃E ∈ I)(ﬁxG(E) ⊆ symG(t))}.

(iii) (Cf. [20, p. 46] and [21, p. 251].) A permutation model (or, equivalently, a Fraenkel-Mostowski model) is every class N which can be defined by (i). Remark 2.10. (a) Let F be a normal filter of subgroups of G and let x ∈M. If symG(x) ∈F, then x is called symmetric. If every element of TC({x}) is symmetric, then x is called hereditarily symmetric (cf. [20, p. 46] and [21, p. 251]). (b) Given a normal ideal I of subsets of the set A of atoms of M, the filter FI of subgroups of G generated by {fixG(E): E ∈ I} is a normal filter such that the permutation model determined by M, G and FI coincides with the permutation model determined by M, G and I (see [20, p. 47]). For x ∈M, a set E ∈ I such that fixG(E) ⊆ symG(x) is called a support of x. In the forthcoming sections, we describe and apply several permutation models. For example, we apply the permutation model which appeared in [26, the proof to Theorem 2.5] and was also used in [27], the Basic Fraenkel Model (labeled as N 1 in [15]) and the Mostowski Linearly Ordered Model (labeled as N 3 in [15]). Let us give definitions of these models and recall some of their properties for future references. Definition 2.11. (Cf. [26].) Let M be a model of ZFA + AC. Let A be the set of all atoms of M and let I = [A]<ω. Assume that:

(i) A is expressed as An where {An : n ∈ N} is a disjoint family such nS∈N that, for every n ∈ N, 1 A = {a : x ∈ S(0, )} n n,x n 1 R2 1 and S(0, n ) is the circle of the Euclidean plane h , ρei of radius n , centered at 0;

15 (ii) G is the group of all permutations of A that rotate the An’s by an angle θn ∈ R .

Then the permutation model Ncr determined by M, G and the normal ideal I will be called the concentric circles permutation model.

Remark 2.12. We need to recall some properties of Ncr for applications in this paper. Let us use the notation from Definition 2.11. In [26, the proof to Theorem 2.5], it was proved that {An : n ∈ N} does not have a multiple choice function in Ncr. In [27, the proof to Theorem 3.5], it was proved that IDI holds in Ncr, so CACfin also holds in Ncr (see Remark 1.10(i)). Definition 2.13. (Cf. [15, p. 176] and [20, Section 4.3].) Let M be a model of ZFA+AC. Let A be the set of all atoms of M and let I = [A]<ω. Assume that: (i) A is a denumerable set; (ii) G is the group of all permutations of A. Then the Basic Fraenkel Model N 1 is the permutation model determined by M, G and I. Remark 2.14. It is known that, in N 1, the set A of all atoms is amorphous, so IDI fails (see [20, p. 52] and [15, pp, 176–177]). It is also known that BPI is false in N 1 but CACfin is true in N 1 (see [15, p. 177]). Definition 2.15. (Cf. [15, p. 182] and [20, Section 4.6].) Let M be a model of ZFA+AC. Let A be the set of all atoms of M and let I = [A]<ω. Assume that: (i) the set A is denumerable and there is a fixed ordering ≤ in A such that hA, ≤i is order isomorphic to the set of all rational numbers equipped with the standard linear order; (ii) G is the group of all order-automorphisms of hA, ≤i. Then the Mostowski Linearly Ordered Model N 3 is the permutation model determined by M, G and I. Remark 2.16. It is known that the power set of the set of all atoms is Dedekind-finite in N 3, so IDI fails in N 3 (see [15, pp. 182–183]). How- ever, BPI and CACfin are true in N 3 (see [15, p, 183]).

16 It is well known that, in any permutation model, the power set of any pure set (that is, a set with no atoms in its transitive closure) is well-orderable (see, e.g., [15, p. 176]). This can be deduced from the following helpful proposition:

Proposition 2.17. (Cf. [20, Item (4.2), p. 47].) Let N be the permutation model determined by M, G and a normal filter F. For every x ∈ N , x is well-orderable in N iff fixG(x) ∈F.

Remark 2.18. If a statement A is satisfied in a permutation model, to show that there exists a ZF-model in which A is satisfied, we use transfer theorems due to Pincus (cf. [37] and [38]). Pincus transfer theorems, together with definitions of a boundable formula and an injectively boundable formula that are involved in the theorems, are included in [15, Note 103]. To our transfer results, we apply mainly the following fragment of the third theorem from [15, p. 286]:

Theorem 2.19. (The Pincus Transfer Theorem.) (Cf. [37], [38] and [15, p. 286].) Let Φ be a conjunction of statements that are either injectively boundable or BPI. If Φ has a permutation model, then Φ has a ZF-model.

In the deﬁnition of an injectively boundable formula, a notion of an injective cardinality is involved. This notion is given, for instance, in [15, Item (3), p. 284]. Let us formulate its equivalent deﬁnition below.

Deﬁnition 2.20. For a set x, the injective cardinality of x is the von Neu- mann cardinal number |x|− deﬁned as follows:

|x|− = sup{κ : κ is a von Neumann cardinal equipotent to a subset of x}.

Now, we are in the position to pass to the main body of the article.

3 New symmetric models

Suppose that Φ is a form that is satisfied in a ZFA-model. Even if Φ fulfills the assumptions of the Pincus Transfer Theorem, it might be complicated to check it and to see well a ZF-model in which Φ is satisfied. This is why it is good to give a direct relatively simple description of a ZF-model satisfying Φ. In the proof to Theorem 3.1 below, we show a class of symmetric models

17 satisfying CH ∧ WO(P(R)) ∧ ¬CACfin. In Section 5, models of this class are applied to a proof that the conjunction Part(R) ∧ ¬M(C, ֒→ [0, 1]R) has a ZF-model (see Theorem 5.13).

Theorem 3.1. Let n, ℓ ∈ ω \{0, 1}. There is a symmetric model Nn,ℓ of ZF such that ℵm Nn,ℓ |= ∀m ∈ n(2 = ℵm+1) ∧ ¬CACℓ. Hence, it is also the case that

Nn,ℓ |= CH ∧ WO(P(R)) ∧ ¬CACfin. Proof. Let us use the terminology and results from [32, Chapter VII] and [20, Chapter 5]. By [32, Theorem 6.18, p. 216], we can fix a countable transitive ℵm model M of ZFC + ∀m ∈ n(2 = ℵm+1). Our plan is to construct a symmetric extension model Nn,ℓ of M with the required properties. Let P = Fn(ω × ℓ × ωn × ωn, 2,ωn) be the set of all partial functions p with |p| < ℵn, dom(p) ⊆ ω × ℓ × ωn × ωn and ran(p) ⊆ 2= {0, 1}, partially ordered by reverse inclusion, i.e., for p, q ∈ P, p ≤ q if and only if p ⊇ q. The poset hP, ≤i has the empty function as its maximum element, which we denote by 1. Furthermore, since ωn is a regular cardinal, it follows from [32, Lemma 6.13, p. 214] that hP, ≤i is an ωn-closed poset. Hence, by [32, Theorem 6.14, p. 214], forcing with P adds no new subsets of ωm for m ∈ n, and hence it adds no new reals or sets of reals, but it does add new subsets of ωn. Furthermore, by [32, Corollary 6.15, p. 215], we have that P preserves cofinalities ≤ ωn, and hence cardinals ≤ ωn. Let G be a P-generic filter over M, and let M[G] be the corresponding generic extension model of M. In view of the above, for every model N with M ⊆ N ⊆ M[G], we have the following:

ℵm N |= ∀m ∈ n(2 = ℵm+1). By [32, Theorem 4.2, p. 201], AC is true M[G]. In M[G], for k ∈ ω, t ∈ ℓ, and i ∈ ωn, we deﬁne the following sets along with their canonical names:

1. ak,t,i = {j ∈ ωn : ∃p ∈ G(p(k, t, i, j)=1)},

ak,t,i = {hˇj,pi : j ∈ ωn ∧ p ∈ P ∧ p(k, t, i, j)=1}.

2. Ak,t = {ak,t,i : i ∈ ωn},

Ak,t = {hak,t,i, 1i : i ∈ ωn}.

18 3. Ak = {Ak,0, Ak,1,...,Ak,(ℓ−1)},

Ak = {hAk,0, 1i, hAk,1, 1i,..., hAk,(ℓ−1), 1i}.

4. A = {Ak : k ∈ ω},

A = {hAk, 1i : k ∈ ω}.

Now, every permutation φ of ω × ℓ × ωn induces an order-automorphism of hP, ≤i by requiring, for every p ∈ P, the following:

dom φ(p)= {hφ(k, t, i), ji : hk, t, i, ji ∈ dom(p)}, (1) φ(p)(φ(k, t, i), j)= p(k, t, i, j).

Let G be the group of all order-automorphisms of hP, ≤i induced (as in (1)) by all those permutations φ of ω × ℓ × ωn which are deﬁned as follows. For every k ∈ ω, let σk be a permutation of ℓ = {0, 1,...,ℓ − 1} and also let ηk be a permutation of ωn. We deﬁne

(2) φ(k, t, i)= hk, σk(t), ηk(i)i, for all hk, t, ii ∈ ω × ℓ × ωn. By (2), it follows that for every φ ∈ G such that φ(k, t, i)= hk, σk(t), ηk(i)i, we have that, for every k ∈ ω and every t ∈ ℓ,

(3) φ(Ak,t)= Ak,σk(t), and thus, for every k ∈ ω,

(4) φ(Ak)= Ak.

It follows that for every φ ∈ G,

(5) φ(A)= A.

For every finite subset E ⊆ ω × ℓ × ωn, we let fixG(E) = {φ ∈ G : ∀e ∈ E(φ(e)= e)} and we also let Γ be the filter of subgroups of G generated <ω by the filter base {fixG(E): E ∈ [ω × ℓ × ωn] }. Then Γ is a normal filter on G (see [20, Section 5.2, p. 64] for the definition of the term “normal filter”). An element x ∈ M is called symmetric if there exists a finite subset E ⊆ ω × ℓ × ωn such that, for every φ ∈ fixG(E), we have φ(x) = x; if such a set E exists, we call E a support of x. An element x ∈ M is called hereditarily symmetric if x and all elements of the transitive closure of x are

19 symmetric. Let HS be the set of all hereditarily symmetric names in M. As in [32, Deﬁnition 2.7, p. 189], for τ ∈ HS, let τG denote the value of the name τ. Let Nn,ℓ = {τG : τ ∈ HS} be the symmetric extension model of M. Then Nn,ℓ ⊂ M[G]. In view of the observations at the beginning of the proof, we have

ℵm Nn,ℓ |= ∀m ∈ n(2 = ℵm+1), and thus Nn,ℓ |= CH ∧ WO(P(R)).

For k ∈ ω, t ∈ ℓ, and i ∈ ωn, the sets ak,t,i, Ak,t, Ak, and A are all elements of Nn,ℓ. Let us fix k ∈ ω, t ∈ ℓ, and i ∈ ωn. Then E = {hk, t, ii} is a support of ak,t,i and Ak,t. By (4) and (5), we have that, for every φ ∈ G, φ(Ak)= Ak and φ(A)= A. Thus, ak,t,i, Ak,t, Ak, and A all belong to Nn,ℓ. For σ, τ ∈ HS, let op(σ, τ) be the name for the ordered pair hσG, τGi (see [32, Definition 2.16, p. 191]). Let f = {hk, Aki : k ∈ ω} and f˙ = {hop(k,ˇ Ak), 1i : k ∈ ω}. Since, for every φ ∈ G, φ(f˙)= f˙, we deduce that f˙ is an HS-name for the mapping f (in M[G]). This proves that A is denumerable in Nn,ℓ. Now, by making suitable adjustments to the proof that CAC2 is false in the Second Cohen Model (see [20, Section 5.4, p. 68]), one may verify that A has no partial choice function in the model Nn,ℓ. We invite interested readers to fill in the missing details.

Remark 3.2. Let us note that Theorem 3.1 provides a class of symmetric models satisfying CH ∧ WO(P(R)) ∧ ¬CACfin.

4 Around ICMDI and M(TB,WO)

Since every compact metric space is totally bounded and every infinite separable Hausdorff space is Dedekind-infinite, let us begin our investigations of the forms of type M(C, ) with a deeper look at the forms M(TB,WO), M(TB,S) and ICMDI. We include a simple proof to the following proposition for completeness.

Proposition 4.1. (ZF) Let X = hX,di is a totally bounded metric space such that X is well-orderable. Then X is separable.

20 Proof. Since X is well-orderable, so is the set Y = (Xn ×{n}). Let ≤ be a nS∈N km ﬁxed well-ordering in Y . For every m ∈ N, let ym = hxm,kmi ∈ X ×{km} 1 be the ﬁrst element of hY, ≤i such that X = {Bd(xm(i), ): i ∈ km}. The S m set D = {xm(i): i ∈ km} is countable and dense in X. mS∈N Theorem 4.2. (ZF)

(i) M(TB,WO) → M(TB,S) and M(C,WO) → M(C,S). None of these implications is reversible.

(ii) M(TB,WO) → M(C,WO) → M(C,S) → ICMDI.

(iii) (Cf. [23, Theorem 7 (i)].) CAC → M(TB,S) → M(C,S).

(iv) Neither M(TB,WO) nor M(TB,S) implies CAC.

Proof. It follows from Proposition 4.1 that the implications from (i) are both true. It is known that, in Feferman’s model M2 in [15], CAC is true but R is not well-orderable (see [15, p. 140]). Then [0, 1] is a compact, metrizable but not well-orderable space in M2. Hence M(TB,S) ∧ ¬M(TB,WO) and M(C,S) ∧ ¬M(C,WO) are both true in M2. This completes the proof to (i). In view of (i), it is obvious that (ii) holds. It is known from [23] that (iii) also holds. It follows from the ﬁrst implication of (i) that to prove (iv), it suﬃces to show that M(TB,WO) does not imply CAC. It was shown in [23, the proof to Theorem 15] that there exists a model M of ZF + ¬CAC in which it is true that if a metric space X = hX,di is sequentially bounded (i.e., every sequence of points of X has a Cauchy’s subsequence), then X is well-orderable and separable. By [23, Theorem 7 (vii)], every totally bounded metric space is sequentially bounded. This shows that exists a model M of ZF in which M(TB,WO) ∧ ¬CAC is true. Hence (iv) holds. That ICMDI does not imply M(C,S) is shown in Proposition 5.1(iv). It is unknown whether M(C,WO) is equivalent to or weaker than M(TB,WO) in ZF. To compare M(C,S) with M(TB,S), we recall that it was proved in [24] that the implication M(TB,S) → CAC(R) holds in ZF; however, the implication CAC → M(TB,S) of Theorem 2.1 is not reversible in ZF. On the other hand, it is known that CAC(R) and M(C,S) are independent of

21 each other in ZF (see, e.g., [23]). The following proposition, together with the fact that M(TB,S) implies M(C,S), shows that M(TB,S) is essentially stronger than M(C,S) in ZF. Proposition 4.3. (ZF) (i) (Cf. [24, Proposition 8].) If every totally bounded metric space is strongly totally bounded, then CAC(R) holds.

(ii) In Cohen’s Original Model M1 of [15] the following hold: M(C,S) is true, M(TB,S) is false and there exists a totally bounded metric space which is not strongly totally bounded.

(iii) M(C,S) does not imply M(TB,S). Proof. That (i) holds was proved in [24]. It is known that BPI is true M1 (see [15, p. 147]). It follows from Theorem 2.7(iii) that M(C,S) holds in M1. On the other hand, it is known that CAC(R) fails in M1 (see [15, p. 147]). Therefore, by (i), it holds in M1 that there exists a totally bounded metric space which is not strongly totally bounded. We recall that a topological space X is called limit point compact if every infinite subset of X has an accumulation point in X (see, e.g., [23]). Proposition 4.4. (ZFA) WoAm implies both M(TB,WO) and “every limit point compact, first-countable T1-space is well-orderable”. Proof. Let us assume WoAm. Consider an arbitrary metric space hX,di. Suppose that X is not well-orderable. By WoAm, there exists an amorphous subset B of X. Let ρ = d ↾ B×B. Lemma 1 of [5] states that every metric on an amorphous set has a finite range. Therefore, the set ran(ρ) = {ρ(x, y): x, y ∈ B} is finite. Since B is infinite, the set ran(ρ) \{0} is non-empty. If ε = min(ran(ρ) \{0}), then there does not exist a finite ε-net in hB, ρi because B is infinite and, for every x ∈ B, Bρ(x, ε)= {x}. This implies that ρ is not totally bounded. Hence d is not totally bounded. Now, suppose that Y = hY, τi is a first-countable, limit point compact T1-space. Let C be an infinite subset of Y . Since Y is limit point compact, the set C has an accumulation point in Y. Let y0 be an accumulation point of C and let {Un : n ∈ N} be a base of neighborhoods of y0 in Y. Since Y is a T1-space, we can inductively define an increasing sequence (nk)k∈N of natural N numbers such that, for every k ∈ , C ∩ (Unk \ Unk+1 ) =6 ∅. This implies that

22 C is not amorphous. Hence, no inﬁnite subset of Y is amorphous, so Y is well-orderable by WoAm.

Corollary 4.5. N 1 |= M(TB,WO).

Proof. This follows from Proposition 4.4 and the known fact that WoAm is true in N 1 (see p. 177 in [15]). To prove that M(TB,WO) does not imply WoAm in ZFA, let us use the model N 3. In what follows, the notation concerning N 3 is the same as in Deﬁnition 2.15. For a, b ∈ A with a < b (where A is the set of atoms of N 3 and ≤ is the ﬁxed linear order on A), we denote by (a, b) the open interval in the linearly ordered set hA, ≤); that is, (a, b) = {x ∈ A : a

Lemma 4.6. (Cf. [18, Lemma 3.17 and its proof].) If X ∈ N 3, E is a support of X and there is x ∈ X for which E is not a support, then there exist a subset Y of X and atoms a, b ∈ A with a < b, such that Y ∈ N 3, E ∩ (a, b) = ∅ and, in N 3, there exists a bijection f : (a, b) → Y having a support E′ such that E ∪{a, b} ⊆ E′ and E′ ∩ (a, b)= ∅.

Theorem 4.7. N 3 |= M(TB,WO).

Proof. We use the notation from Definition 2.15. Suppose that hX,di is a metric space in N 3 such that X is not well-orderable in N 3. Then X is infinite. Let E ∈ [A]<ω be a support of both X and d. By Proposition 2.17, there exists x ∈ X such that E is not a support of x. By Lemma 4.6, there exist a, b ∈ A with a < b and (a, b) ∩ E = ∅, such that there exists in N 3 an injection f : (a, b) → X which has a support E′ such that E ∪{a, b} ⊆ E′ and E′ ∩ (a, b) = ∅. We put B = (a, b) and ρ(x, y)= d(f(x), f(y)) for all x, y ∈ B. Let us notice that ρ ∈N 3 because E′ is also a support of ρ. We prove that ran(ρ)= {ρ(x, y): x, y ∈ B} is a two- element set. To this aim, we fix b1, b2 ∈ B with b1 < b2 and put r = ρ(b1, b2). Let u, v ∈ B and u =6 v. To show that ρ(u, v)= r, we must consider several cases regarding the ordering of the elements b1, b2,u,v. We consider only one of the possible cases since all the other cases can be treated in much the same way as the chosen one. So, assume, for example, that b2

23 r = ρ(b1, b2) = ρ(φ(b1),φ(b2)) = ρ(u, v). Therefore, ran(ρ) = {0,r}. Since the range of ρ is ﬁnite, in much the same way, as in the proof to Proposition 4.4, we deduce that hB, ρi is not totally bounded. Hence d is not totally bounded. Corollary 4.8. M(TB,WO) does not imply WoAm in ZFA. Proof. It is known that WoAm is false in N 3 (see [15, p.183]). Therefore, the conjunction M(TB,WO) ∧ ¬WoAm is true in N 3 by Theorem 4.7. In contrast to Corollary 4.5 and Theorem 4.7, we have the following proposition:

Proposition 4.9. Ncr |= ¬M(C,WO).

Proof. It was shown in [26, the proof to Theorem 2.5] that, in Ncr, there exists a compact metric space X = hX,di which is not weakly Loeb. Then X cannot be well-orderable in Ncr. Remark 4.10. In [40], a symmetric model M of ZF was constructed such that, in M, there exists a compact metric space hX,di which is not weakly Loeb; thus, M(C,WO) fails in M. It is obvious that IDI implies ICMDI in ZF; however, it seems to be still an open problem of whether this implication is not reversible in ZF. To solve this problem, ﬁrst of all, let us notice that the following corollary follows directly from Corollary 4.5 and Theorem 4.7: Corollary 4.11. (i) N 1 |=(ICMDI ∧ ¬IDI).

(ii) N 3 |=(BPI ∧ ICMDI ∧ ¬IDI). To transfer BPI ∧ ICMDI ∧ ¬IDI to a model of ZF, let us prove the following lemma: Lemma 4.12. ICMDI is injectively boundable. Proof. First, we put

Φ(x)= ¬(∃y)(y ⊆ x ∧|y| = ω), and Ψ(x)= “x is ﬁnite”.

24 Note that the formula Ψ(x) is boundable (see [37] or Note 103, p. 284 in [15]). Now it is not hard to verify that ICMDI is logically equivalent to the statement Ω, where

Ω=(∀x)(|x|− ≤ ω → (∀ρ ∈ P(x × x × R)) [(Φ(x) ∧ (ρ is a metric on x such that hX, ρi is compact)) → Ψ(x)])

Since Ω is obviously injectively boundable, so is ICMDI. Theorem 4.13. The conjunction BPI ∧ ICMDI ∧ ¬IDI has a ZF-model. Proof. It is known that ¬IDI is boundable, and hence injectively boundable (see [37, 2A5, p. 772] or [15, p. 285]). By Theorem 2.19, if a conjunction of BPI with injectively boundable statements has a Fraenkel-Mostowski model, then it has a ZF-model. This, together with Corollary 4.11(ii) and Lemma 4.12, completes the proof. Theorem 4.14. The conjunction (¬BPI)∧ICMDI∧¬IDI has a ZF-model. Proof. Let Φ be the conjunction (¬BPI)∧ICMDI∧¬IDI. It is known that BPI is false in N 1 (see [15, p. 177]). Hence Φ has a permutation model by Corollary 4.11(i). Since the statements ¬BPI, ICMDI and ¬IDI are all injectively boundable, Φ has a ZF-model by Theorem 2.19. Corollary 4.15. ICMDI does not imply IDI in ZF. Furthermore, BPI is independent of ZF + ICMDI + ¬IDI.

Proposition 4.16. ICMDI implies CACfin in ZF. Proof. It suffices to apply Corollary 2.2 (iii) in [27] which states that if every infinite compact metrizable space has an infinite well-orderable subset, then CACfin holds. At this moment, it is unknown whether there is a model of ZF in which the conjunction CACfin ∧ ¬ICMDI is false. Remark 4.17. In Cohen’s original model M1 in [15], CUC holds and there exists a dense Dedekind-finite subset X of the interval [0, 1] of R. The metric space X = hX,di, where d(x, y)= |x − y| for all x, y ∈ X, is totally bounded but X is not well-orderable in M1. It was remarked in [24] that, since hX,di is not separable, CUC does not imply M(TB,S) in ZF. Now, it is easily seen that CUC does not imply M(TB,WO) in ZF.

25 It is not stated in [15] nor in [16] that WoAm implies CUC. Since we have not seen a solution of the problem of whether this implication is true in other sources, let us notice that it follows from the following proposition that this implication holds in ZF: Proposition 4.18. (i) (ZFA) WoAm → CUC; (ii) If N is a model of ZFA in which R is well-orderable (in particular, if N is a permutation model), then:

N |=(CACWO → CUC).

(iii) If N is a permutation model, then:

N |=(ACfin → CUC).

Proof. Let A = {An : n ∈ ω} be a disjoint family of non-empty countable sets and let A = An. Clearly, if (An × ω) is countable, then A is nS∈ω nS∈ω countable. Therefore, to show that A is countable, we may assume that, for every n ∈ ω, the set An is denumerable. For every n ∈ ω, let Bn be the set ω ω of all bijections from ω onto An. Since |ω | = |R | = |R| and the sets An are all denumerable, for every n ∈ ω, the set Bn is equipotent to R. (i) Let B = Bn. If B is well-orderable, then there exists a sequence nS∈ω (fn)n∈ω of bijections fn : ω → An, so A is countable. Suppose that B is not well-orderable. Then it follows from WoAm that there exists an amorphous subset C of B. Since C cannot be partitioned into two inﬁnite subsets, the set {n ∈ ω : C ∩ Bn =6 ∅} is ﬁnite. This implies that there exists m ∈ ω such that C ⊆ Bn, so C is equipotent to a subset of R. But this is n∈Sm+1 impossible because R does not have amorphous subsets. The contradiction obtained completes the proof to (i).

(ii) Now, assume that R is well-orderable and CACWO holds. Then the sets Bn, being equipotent to R, are all well-orderable. Hence, it follows from CACWO that there exists f ∈ Bn. Then we have a sequence (f(n))n∈ω nQ∈ω of bijections f(n): ω → An, so A is countable. (iii) Since ACfin implies ACWO in every permutation model (cf. [13] and Note 2 in [15]), we infer that if ACfin is satisﬁed in N and N is a permutation model, then CACWO holds in N . This, taken together with (ii), implies (iii).

26 Remark 4.19. (a) In Felgner’s Model I (labeled as M20 in [15]), ACWO holds and CUC fails (cf. [15, p. 159]). We recall that ACfin ∧ ¬ACWO is true in Sageev’s Model I (labeled as model M6 in [15]). Moreover, since IDI∧¬CUC is true in M6 (cf. [15, p. 152]), ICMDI does not imply CUC is ZF. (b) One should not claim that CACfin and CACWO are equivalent in every permutation model. Namely, let N be the permutation model of IDI∧ ¬CUC constructed in [41, the proof to Theorem 4 (iv)]. Since IDI implies CACfin, it follows from Proposition 4.18 that, in this model N , CACfin is true but CACWO is false. Remark 4.20. (a) It is unknown whether M(C,S) implies CUC in ZF or in ZFA. We recall that BPI implies M(C,S). The problem of whether BPI implies CUC in ZF or in ZFA is still unsolved. However, if N is a permutation model in which BPI is true, then ACfin is also true in N (see, e.g., [12, Proposition 4.39]); hence, by Proposition 4.18(iii), BPI implies CUC in every permutation model. (b) Since M(C,S) implies CACfin (see Theorem 2.2(ii) or Theorem 4.2 with Proposition 4.16), it follows from Proposition 4.18(iii) that CACfin ∧ ¬ACfin is satisﬁed in every permutation model of M(C,S) ∧ ¬CUC. It still eludes us whether or not CUC implies M(C,S) in ZF. However, we are able to provide a partial solution to this intriguing open problem by proving that the statement UT(ℵ0, ℵ0,cuf) ∧ ¬ICMDI is a conjunction of injectively boundable statement and it has a permutation model. It is obvious that ¬ICMDI is boundable, so also injectively boundable. To show that UT(ℵ0, ℵ0,cuf) is injectively boundable, we need the following lemma proved in [17]: Lemma 4.21. (ZF) (Cf. [17, Lemma 3.5].) For any ordinal α, if R is a collection of sets such that |R| ≤ ℵα+1 and, for every x ∈ R, |x|≤ℵα, then | R| 6≥ ℵα+2. S Proposition 4.22. The statement UT(ℵ0, ℵ0,cuf) is injectively boundable.

Proof. In the light of Lemma 4.21, UT(ℵ0, ℵ0,cuf) is equivalent to the statement:

(6) (∀x)(|x| 6≥ ℵ3 → (∀y) “if y is a countable collection of countable sets whose union is x, then x is a cuf set”).

27 Since, for every set x, the statements |x| 6≥ ℵ3 and |x|− ≤ℵ2 are equivalent, it is obvious that (6) is injectively boundable. Thus, UT(ℵ0, ℵ0,cuf) is also injectively boundable.

Theorem 4.23. (i) The statement LW∧UT(ℵ0, ℵ0,cuf)∧¬ICMDI has a permutation model.

(ii) The statement UT(ℵ0, ℵ0,cuf) ∧ ¬ICMDI has a ZF-model.

Proof. (i) Let us apply the permutation model N which was constructed in [6, the proof to Theorem 3.3]. To describe N , we start with a model M of ZFA + AC with a set A of atoms such that A has a denumerable partition {Ai : i ∈ ω} into denumerable sets, and for each i ∈ ω, Ai has a denumerable partition Pi = {Ai,j : j ∈ N} into ﬁnite sets such that, for every j ∈ ω \{0}, |Ai,j| = j. Let Sym(A) be the group of all permutations of A and let

G = {φ ∈ Sym(A):(∀i ∈ ω)(φ(Ai)= Ai) and |{x ∈ A : φ(x) =6 x}| < ℵ0}.

Let Pi = {φ(Pi): φ ∈ G} and also let P = {Pi : i ∈ ω}. Let F be the S normal filter of subgroups of G generated by the collection {fixG(E): E ∈ [P]<ω}. Then N is the permutation model determined by M, G and F. We say that a finite subset E of P is a support of an element x of N if, for every φ ∈ fixG(E), φ(x)= x. It was noticed in [6, the proof to Theorem 3.3] that, for every i ∈ ω and every Q ∈ Pi, the following hold:

(a) for any φ ∈ G, φ ﬁxes Q if and only if φ ﬁxes Q pointwise;

(b) (∃jQ ∈ ω)(Q ⊇{Ai,j : j > jQ}).

To prove that LW is true in N , we fix a linearly ordered set hY, ≤i in N <ω and prove that fixG(Y ) ∈ F. To this aim, we choose a set E ∈ [P] such that E is a support of both Y and ≤. To show that fixG(E) ⊆ fixG(Y ), let us consider any element y ∈ Y and a permutation φ ∈ fixG(E). Suppose that φ(y) =6 y. Then either y<φ(y) or φ(y) < y. Since every element of G moves only finitely many atoms, there exists k ∈ ω \{0} such that φk is the indentity mapping on A. Assuming that φ(y) < y, for such a k, we obtain the following:

y<φ(y) <φ2(y) <...<φk−1(y) <φk(y)= y,

28 and thus y

A0,n0 ∈ φi(P0) for all i ∈ n +1.

Assume that f(A0,n0 )= x0. Since |A0,n0 | = n0 ≥ 2, there exists y0 ∈ A0,n0 such that y0 =6 x0. Let η = (x0,y0), i.e., η is the permutation of A which interchanges x0 and y0, and ﬁxes all other atoms of N . Then η(A0,n0 ) = ′ ′ An0 and η ∈ ﬁxG(D ). Since D is a support of f, we have η(f) = f.

Therefore, since hA0,n0 , x0i ∈ f, we infer that hη(A0,n0 ), η(x0)i ∈ η(f) = f, so hA0,n0 ,y0i ∈ f and, in consequence, x0 = y0. The contradiction obtained shows that A does not have a partial choice function in N . This implies that the set A0 is Dedekind-ﬁnite, and, thus, ICMDI is false in N . (ii) Let Φ be the statement UT(ℵ0, ℵ0,cuf) ∧ ¬ICMDI. We have al- ready noticed that ¬ICMDI is injectively boundable. This, together with Proposition 4.22, implies that Φ is a conjunction of injectively boundable statements. Therefore, (ii) follows from (i) and from Theorem 2.19.

Clearly, every ZF- model for UT(ℵ0, ℵ0,cuf) ∧ ¬ICMDI is also a model for UT(ℵ0, ℵ0,cuf) ∧ ¬M(C,S). This, together with Theorem 4.23, implies the following corollary:

29 Corollary 4.24. The conjunction UT(ℵ0, ℵ0,cuf) ∧ ¬M(C,S) has a ZF- model.

Remark 4.25. Let N be the model that we have used in the proof to Theorem 4.23. The proof to Theorem 3.3 in [6] shows that UT(ℵ0,cuf,cuf) is false in N . Hence CUC is also false in N . Since the statement UT(ℵ0, ℵ0,cuf) ∧ ¬UT(ℵ0,cuf,cuf) has a permutation model (for instance, N ), it also has a ZF-model by Proposition 4.22 and Theorem 2.19.

5 The forms of type M(C, )

It is known that every separable metrizable space is second-countable in ZF. It is also known, for instance, from Theorem 4.54 of [12] or from [9] that, in ZF, the statement “every second-countable metrizable space is separable” is equivalent to CAC(R). The negation of CAC(R) is relatively consistent with ZF, so it is relatively consistent with ZF that there are non-separable second-countable metrizable spaces. On the other hand, by Theorem 2.7(i), it holds in ZF that separability and second-countability are equivalent in the class of compact metrizable spaces. Theorem 2.1 shows that totally bounded metric spaces are second-countable in ZF + CAC; in particular, it holds in ZF+CAC that all compact metrizable spaces are second-countable. However, the situation is completely diﬀerent in ZF. There exist ZF-models including compact non-separable metric spaces. Namely, it follows from The- orem 2.2 that in every ZF-model satisfying the negation of CACfin, there exists an uncountable, non-separable compact metric space whose size is in- comparable to |R|. The following proposition shows (among other facts) that M(C, ≤ |R|) and M(C,S) are essentially stronger than CACfin in ZF and, furthermore, IDI is independent of both ZF + M(C, ≤ |R|) and ZF + M(C,S).

Proposition 5.1. (i) (ZFA) M(C,S) → M(C, ≤|R|) → CACfin.

(ii) Ncr |=(¬M(C,S)) ∧ ¬M(C, ≤|R|).

(iii) CACfin implies neither M(C, ≤|R|) nor M(C,S) in ZF.

(iv) IDI implies neither M(C, ≤|R|) nor M(C,S) in ZF.

(v) Neither M(C, ≤|R|) nor M(C,S) implies IDI in ZF.

30 Proof. (i) It follows directly from Theorem 2.2 that the implications given in (i) are true in ZF; however, the arguments from [23] are sufficient to show that these implications are also true in ZFA. (ii) By Proposition 4.9, there exists a compact metric space X = hX,di in Ncr such that the set X is not well-orderable in Ncr. Since R is well- orderable in Ncr, the set X is not equipotent to a subset of R in Ncr. Hence M(C, ≤|R|) fails in Ncr. This, together with (i), implies (ii). (iii)–(iv) Let Φ be either CACfin or IDI. In the light of (i), to prove (iii) and (iv), it suffices to show that the conjunction Φ ∧ ¬M(C, ≤|R|) has a ZF-model. It follows from (ii) that the conjunction Φ ∧ ¬M(C, ≤|R|) has a permutation model (for instance, Ncr). Therefore, since the statements CACfin, IDI and ¬M(C, ≤|R|) are all injectively boundable, Φ∧¬M(C, ≤ |R|) has a ZF-model by Theorem 2.19. (v) Let Ψ be either M(C, ≤ |R|) or M(C,S). Since BPI is true in N 3, it follows from Theorem 2.7 (iii) that Ψ is true in N 3. It is known that IDI is false in N 3. Hence, the conjunction Ψ ∧ ¬IDI has a permutation model. To complete the proof, it suffices to apply Theorem 2.19.

Theorem 5.2. (ZF)

(i) (CACfin ∧ M(C, σ − l.f)) ↔ M(C,S).

(ii) (CACfin ∧ M(C,STB)) ↔ M(C,S).

Proof. Let X = hX,di be a compact metric space. (→) We assume both CACfin and M(C, σ − l.f). By our hypothesis, X has a base B = {Bn : n ∈ N} such that, for every n ∈ N, the family S Bn is locally finite. In ZF, to check that if A is a locally finite family in X, then it follows from the compactness of X that A is finite, we notice that the collection V of all open sets V of X such that V meets only finitely many members of A is an open cover of X, so V has a finite subcover. In consequence, X meets only finitely many members of A, so A is finite. Therefore, for every n ∈ N, the family Bn is finite. This, together with CACfin, implies that the family B is countable, so X is second-countable. Hence, by Theorem 2.7(i), X is separable as required.

(←) By Proposition 5.1, M(C,S) implies CACfin. To conclude the proof to (i), it suﬃces to notice that M(C,S) implies M(C, 2) and M(C, 2) trivially implies that every compact metric space has a σ - locally ﬁnite base.

31 (ii) (→) Now, we assume both CACfin and M(C,STB). Since X is strongly totally bounded, it follows that it admits a sequence (Dn)n∈N such N 1 X CAC that, for every n ∈ , Dn is a n -net of . By fin, the set D = {Dn : n ∈ N} is countable. Since, D is dense in X, it follows that X is separable.S (←) It is straightforward to check that every separable compact metric space is strongly totally bounded. Hence M(C,S) implies M(C,STB). Proposition 5.1 completes the proof. Remark 5.3. By Proposition 5.1, there exists a model M of ZF in which CACfin holds and M(C,S) fails. By Theorem 5.2(i), M(C, σ − l.f) fails in M. This, together with Theorem 2.5(i), implies that M(C, σ − l.f.) independent of ZF. Theorem 5.4. (ZF) (i) (CAC(R,C) ∧ M(C, ≤|R|)) ↔ M(C,S).

(ii) CAC(R) does not imply M(C, ≤|R|). Proof. (i) (→) We assume CAC(R,C) and M(C, ≤|R|). We ﬁx a compact metric space X = hX,di and prove that X is separable. For every n ∈ N, let n n n X = hX ,dni where dn is the metric on X deﬁned by:

dn(x, y) = max{d(x(i),y(i)) : i ∈ n}. Then Xn is compact for every n ∈ N. By M(C, ≤|R|), |X|≤|R|. Therefore, N since |X |≤|R|, there exists a family {ψn : n ∈ N} such that, for every n n ∈ N, ψn : X → R is an injection. The metric d is totally bounded, so, for every n ∈ N, the set 1 M = {m ∈ N : ∃y ∈ Xm, ∀x ∈ X,d(x, {y(i): i ∈ n}) < } n n is non-empty. Let kn = min Mn for every n ∈ N. To prove that X is strongly totally bounded, for every n ∈ N, we consider the set Cn deﬁned as follows: 1 C = {y ∈ Xkn : ∀x ∈ X(d(x, {y(i): i ∈ k }) < )}. n n n

kn We claim that for every n ∈ N,Cn is a closed subset of X . To this end, kn we ﬁx y0 ∈ X \ Cn. Then, since X is inﬁnite, there exists x0 ∈ X such 1 that Bd(x0, n ) ∩{y0(i): i ∈ n} = ∅. Let r = d(x0, {y0(i): i ∈ n}) and

32 1 ε = r − n . Then ε > 0. To show that Bdkn (y0,ε) ∩ Cn = ∅, suppose that z0 ∈ Bdkn (y0,ε) ∩ Cn. Then 1 d(x , {z (i): i ∈ k }) = max{d(x , z (i)) : i ∈ k } < 0 0 n 0 0 n n and

dkn (y0, z0) = max{d(y0(i), z0(i)) : i ∈ kn} < ε.

For every i ∈ kn, we have:

r ≤ d(x0,y0(i)) ≤ d(x0, z0(i)) + d(z0(i),y0(i)) ≤ d(x0, z0(i)) + dkn (z0,y0).

Hence, for every i ∈ kn, the following inequalities hold: 1 r − d n (z ,y ) ≤ d(x , z (i)) < . k 0 0 0 0 n

In consequence, ε

33 Corollary 5.5. In every permutation model, M(C, ≤|R|) and M(C,S) are equivalent. Proof. Let N be a permutation model. Since R is well-orderable in N (see Subsection 2.4), CAC(R) is true in N . This, together with Theorem 5.4(i), completes the proof. Remark 5.6. (i) The proof to Corollary 5.5 shows that M(C, ≤ |R|) and M(C,S) are equivalent in every model of ZFA in which R is well-orderable. (ii) In much the same way, as in the proof to Theorem 5.4(ii)(←), one can show that, for every family {Xn : n ∈ ω} of pairwise disjoint compact spaces, if the direct sum X = Xn is metrizable, then it is separable. nL∈ω We include a sketch of a ZF-proof to the following lemma for complete- (ness. We use this lemma in our ZF-proof that M(C, ֒→ [0, 1]N) and M(C,S are equivalent. Lemma 5.7. (ZF) Suppose that B is a base of a non-empty metrizable space X = hX, τi. Then there exists a homeomorphic embedding of X into the cube [0, 1]B×B. Proof. We may assume that X consists of at least two points. Let d be a metric on X such that τ = τ(d) and let

W = {hU, V i∈B×B : ∅= 6 clXU ⊆ V =6 X}.

For every W = hU, V i ∈ W, by deﬁning

d(x, clX(U)) fW (x)= whenever x ∈ X, d(x, clX(U)) + d(x, X \ V ) we obtain a continous function from X into [0, 1]. Let h : X → [0, 1]W be the evaluation mapping deﬁned by: h(x)(W )= fW (x) for all x ∈ X and W ∈ W. Then h is a homeomorphic embedding of X into [0, 1]W . To complete the proof, it suﬃces to notice that [0, 1]W is homeomorphic to a subspace of [0, 1]B×B. Theorem 5.8. (ZF) .(i) M(C, ֒→ [0, 1]N) ↔ M(C,S)

.(ii) M(C, ≤|R|) → M(C, ֒→ [0, 1]R)

34 Proof. Let X = hX, τi be an inﬁnite compact metrizable space and let d be a metric on X such that τ = τ(d). i) (→) We assume M(C, ֒→ [0, 1]N) and show that X is separable. By) our hypothesis, X is homeomorphic to a compact subspace Y of the Hilbert cube [0, 1]N. Since [0, 1]N is second-countable, it follows from Theorem 2.7(i) (that Y is separable. Hence X is separable. In consequence, M(C, ֒→ [0, 1]N implies M(C,S). (←) If M(C,S) holds, then every compact metrizable space is second- countable. Since, by Lemma 5.7, every second-countable metrizable space is .(embeddable in the Hilbert cube [0, 1]N, M(C,S) implies M(C, ֒→ [0, 1]N (ii) Now, suppose that M(C, ≤ |R|) holds. Then |X|≤ |R|. Since |[R]<ω| = |R|, we infer that |[X]<ω|≤|R|. For every n ∈ N, let

1 k = min{m ∈ N : B (x, )= X for some A ∈ [X]m} n [ d n x∈A and 1 E = {A ∈ [X]km : B (x, )= X}. n [ d n x∈A 1 N Let B = {Bd(x, n ): x ∈ A, A ∈ En, n ∈ }. It is straightforward to verify that B is a base for X of size |B| ≤ |R × N|≤|R|, so |B×B| ≤|R|. This, together with Lemma 5.7, implies that X is embeddable into [0, 1]R. Hence .(M(C, ≤|R|) implies M(C, ֒→ [0, 1]R

In view of Theorem 5.8, one may ask the following questions: ?(Question 5.9. (i) Does M(C, ֒→ [0, 1]R) imply M(C, ֒→ [0, 1]N

?(|ii) Does M(C, ֒→ [0, 1]R) imply M(C, ≤|R)

R ?( [iii) Does CACfin imply M(C, ֒→ [0, 1)

R ?iv) Does M(C, ֒→ [0, 1] ) imply CACfin) Remark 5.10. (a) Regarding Question 5.9 (i)-(ii), we notice that the answer is in the aﬃrmative in permutation models. Indeed, let N be a permutation model. It is known that R and P(R) are well-orderable in N (see Subsection Therefore, assuming that M(C, ֒→ [0, 1]R) holds in N and working .(2.4 inside N , we deduce that, given a compact metrizable space X in N , X embeds in [0, 1]R. Hence X is a well-orderable space, so X is Loeb. Since

35 X is a compact metrizable Loeb space, by Theorem 2.7(i), X is second- countable. Therefore, by Lemma 5.7, X embeds in [0, 1]N and, consequently, |X|≤|R|. (b) Regarding Question 5.9(iii), we note that CACfin holds in the per- R mutation model Ncr. To show that M(C, ֒→ [0, 1] ) fails in Ncr, we notice that, by Proposition 4.9, there exists a compact metric space X = hX,di in R Ncr such that X is not well-orderable in Ncr. Since [0, 1] , being equipotent to the well-orderable set P(R) of Ncr, is well-orderable in Ncr, it is true in R R ( [Ncr that X is not embeddable in [0, 1] . This explains why M(C, ֒→ [0, 1 R ( [fails in Ncr. Therefore, since the conjunction CACfin ∧ ¬M(C, ֒→ [0, 1 has a permutation model, it also has a ZF-model by Theorem 2.19. To give more light to Questions 5.9 (iii)-(iv), let us prove the following Theorems 5.11 and 5.12. Theorem 5.11. (ZF)

(i) M(C, |BY | ≤ |B|) → M([0, 1], |BY | ≤ |B|) → IDI(R). (ii) The following are equivalent: (a) Every compact (0-dimesional) subspace of the Tychonoff cube [0, 1]R has a base of size ≤|R|; (b) every compact (0-dimensional) subspace of the Tychonoff cube [0, 1]R with a unique accumulation point has a base of size ≤|R|; (c) Part(R). (iii) Every compact metrizable subspace of the Tychonoff cube [0, 1]R with a unique accumulation point has a base of size ≤ |R| iff for every denumerable family A of finite subsets of P(R) such that A is pairwise disjoint, | A|≤ |R|. S S Proof. (i) Assume that IDI(R) is false. By a well-known result of N. Brunner (cf. [15, Form [13 A]]), there exists a Dedekind-finite dense subset of the interval (0, 1) with its usual topology. Then B = {(x, y): x, y ∈ D,x

X = {f}∪{fP : P ∈ P}.

We claim that the subspace X of [0, 1]R is compact. To see this, let us consider an arbitrary family U of open subsets of [0, 1]R such that X ⊆ U. There S exists U0 ∈ U such that f ∈ U0. There exist ε ∈ (0, 1) and a non-empty ﬁnite subset J of R such that the set

V = {π−1([0,ε)) : j ∈ J} \ j R is a subset of U0 where, for each j ∈ R and x ∈ [0, 1] , πj(x) = x(j). Since J is finite, there exists a finite set PJ ⊆ P such that J ⊆ PJ . We notice S that, for every P ∈P\PJ and every j ∈ J, fP (j)=0. Hence fP ∈ U0 for every P ∈P\PJ . This implies that there exists a finite set W such that W ⊆ U and X ⊆ W. Hence X is compact as claimed. If P ∈ P and j ∈ P , −1 1 S X then πj (( 2 , 1]) ∩ X = {fP }, so fP is an isolated point of . Hence f is the unique accumulation point of X. The space X is also 0-dimensional. By (b), X has a base B equipotent to a subset of R. Since {{fP } : P ∈P}⊆B, it follows that |P|≤ |R| as required. (c) → (a) We assume Part(R) and fix a compact subspace X of the cube [0, 1]R. It is well known that [0, 1]R is separable in ZF (cf., e.g., [25]). Fix a countable dense subset D of [0, 1]R. For every y ∈ D, let

V = { {π−1((y(i) − 1/m,y(i)+1/m)) : i ∈ F } : ∅= 6 F ∈ [R]<ω, m ∈ N}. y \ i <ω Since |[R] | = |R × N| = |R| in ZF, it follows that B = {Vy : y ∈ D} is equipotent to R. It is a routine work to verify that B isS a base for [0, 1]R. Deﬁne an equivalence relation ∼ on B by requiring:

(8) O ∼ Q iﬀ O ∩ X = Q ∩ X.

37 Clearly

(9) BX = {P ∩ X :[P ] ∈ B/ ∼} is a base for X of size |B/ ∼|. Since |B/ ∼|≤|R|, it follows that |BX |≤|R| as required.

(iii) (→) Fix family A = {An : n ∈ N} of finite subsets of P(R) such that the family P0 = A is pairwise disjoint. Let P1 = P0 ∪{R \ P0} S S and P = P1 \ {∅}. Then P is a partition of R. Let f, fP with P ∈ P and X be defined as in the proof of (ii) that (b) implies (c). Since P is a cuf set, the space X has a σ-locally finite base. This, together with Theorem 2.5(ii), implies that X is metrizable. Suppose X has a base B such that |B| ≤ |R|. In much the same way, as in the proof that (b) implies (c) in (ii), we can show that |P|≤ |R|. Then | A|≤|R|. S (←) Now, we consider an arbitrary compact metrizable subspace X of R the cube [0, 1] such that X has a unique accumulation point. Let x0 be the accumulation point of X and let d be a metric on X which induces the topology of X. For every x ∈ X\{x0} let 1 n = min{n ∈ N : B (x, )= {x}}. x d n For every n ∈ N, let En = {x ∈ X : nx = n}.

Without loss of generality, we may assume that, for every n ∈ N, En =6 ∅. Since X is compact, it follows easily that, for every n ∈ N, the set En is ﬁnite. Let us apply the base B of [0, 1]R given in the proof of part (ii) that (c) implies (a). Let ∼ be the equivalence relation on B given by (8). Let

CX = {[P ] ∈ B/ ∼ : |P ∩ (X \{x0})| =1}. For every n ∈ N, let

Cn = {[P ] ∈ CX : P ∩ En =6 ∅}.

Clearly, for every n ∈ N, |Cn| = |En|. Let C = {Cn : n ∈ N}. There S exists a bijection ψ : B → R. For every n ∈ N. we put An = {{ψ(U): U ∈ H} : H ∈ Cn}. Then, for every n ∈ N, An is a ﬁnite subset of P(R). Let A = {An : n ∈ N}. Then A is pairwise disjoint. Suppose that | A|≤|R|. S S Then |C| ≤ |R|. This implies that the family W = {P ∩ (X \{x0}):[P ] ∈ C} R 1 N X is of size ≤| |. The family G = W∪{Bd(x0, n ): n ∈ } is a base of such that |G| ≤ |R|.

38 The following theorem leads to a partial answer to Question 5.9(iv).

Theorem 5.12. (ZF)

.((i) (M(C, ֒→ [0, 1]R) ∧ Part(R)) ↔ M(C, B(R)

(ii) M(C,S) → M(C, W (R)) → M(C, B(R)) → CACfin.

(iii) (CAC(R) ∧ M(C, B(R)) → M(C,S).

(iv) CAC(R) → (M(C,S) ↔ M(C, W (R)) ↔ M(C, B(R)).

Proof. (i) This follows from Theorem 5.11(ii) and Lemma 5.7. (ii) It is obvious that M(C, W (R)) implies M(C, B(R)). Assume M(C,S) and let Y = hY, τi be an infinite compact metrizable separable space. Since Y is second-countable and |Rω| = |R|, it follows that |τ|≤|R|. To show that |R|≤|τ|, we notice that, since X is infinite and X is second-countable, there exists a disjoint family {Un : n ∈ ω} such that, for each n ∈ ω, Un ∈ τ. For J ∈ P(ω), we put ψ(J) = {Un : n ∈ J} to obtain an injection ψ : ω → τ. Hence |R| = |P(ω)|≤|τ|. S To see that M(C, B(R)) → CACfin, we assume M(C, B(R)), fix a disjoint family A = {An : n ∈ N} of non-empty finite sets and show that A has a choice function. To this aim, we put A = A, take an element ∞ ∈/ A and S X = A ∪ {∞}. For each n ∈ N, let ρn be the discrete metric on An. Let d be the metric on X defined by (∗) in Subsection 2.3. By our hypothesis, the space X = hX,di has a base B of size ≤|R|. Let ψ : B→ R be an injection. Since {{x} : x ∈ A} ⊆ B and the sets An are finite, for each n ∈ N, we can ⋆ ⋆ ⋆ N define An = {ψ({x}): x ∈ An} and an = min An. For each n ∈ , there is ⋆ a unique xn ∈ An such that ψ({xn}) = an. This shows that A has a choice function. (iii) Now, we assume both CAC(R) and M(C, B(R)). Let us consider an arbitrary compact metric space X = hX, ρi. By our hypothesis, X has a base B of size ≤|R|. Since, |[R]<ω|≤|R|, it follows that |[B]<ω|≤|R|. For every n ∈ N, let 1 A = {F ∈ [B]<ω : F = X ∧ ∀F ∈F(δ (F ) ≤ )}. n [ ρ n

Since X is compact, ρ is totally bounded. Therefore, An =6 ∅ for every n ∈ N. By CAC(R), we can ﬁx a sequence (Fn)n∈N such that, for every

39 <ω n ∈ N, Fn ∈ An. Since |[B] |≤|R|, we can also ﬁx a sequence (≤n)n∈N such that, for every n ∈ N, ≤n is a well-ordering on Fn. This implies that the family F0 = {Fn : n ∈ N} is countable. Furthermore, it is a routine work S to verify that F0 is a base of X. Hence X is second-countable. By Theorem 2.7 (i), X is separable. This completes the proof to (iii). That (iv) holds follows directly from (ii) and (iii). Our proof to the following theorem emphasizes the usefulness of Theorems 3.1 and 5.12: Theorem 5.13. (a) The following implications are true in ZF:

R .(WO(P(R)) → WO(R) → Part(R) → (M(C, ֒→ [0, 1] ) → CACfin

(b) There exists a symmetric model of ZF + CH + WO(P(R)) in which M(C, ֒→ [0, 1]R) is false. Hence, M(C, ֒→ [0, 1]R) does not follow from Part(R) in ZF. Proof. It is obvious that the first two implications of (a) are true in ZF. Thus, it follows directly from Theorem 5.12 (i)–(ii) that (a) holds. To prove (b), let us notice that, in the light of Theorem 3.1, we can fix a symmetric model M of ZF + CH + WO(P(R)) + ¬CACfin. It follows from (a) that .Part(R) is true in M but M(C, ֒→ [0, 1]R) fails in M Remark 5.14. (i) To show that Part(R) is not provable in ZF, let us recall that, in [11], a ZF-model Γ was constructed such that, in Γ, there exists a family F = {Fn : n ∈ N} of two-element sets such that F is a partition of R but F does not have a choice function. Then, in Γ,S there does not exist an injection ψ : F → R (otherwise, F would have a choice function in Γ). Hence Part(R) Sfails in Γ. (ii) Since Part(R) is independent of ZF, it follows from Theorem 5.11(ii) that it is not provable in ZF that every compact metrizable subspace of the (cube [0, 1]R has a base of size ≤ |R|. We do not know if M(C, ֒→ [0, 1]R implies every compact metrizable subspace of the cube [0, 1]R has a base of size ≤|R|. (iii) It is not provable in ZFA that every compact metrizable space with a unique accumulation point embeds in [0, 1]R. Indeed, in the Second Fraenkel model N 2 of [15], there exists a disjoint family of two-element sets A = {An : n ∈ N} whose union has no denumerable subset. Let A = A, ∞ ∈/ A, S 40 X = A ∪{∞} and, for every n ∈ N, let ρn be the discrete metric on An. Let d be the metric on X defined by (∗) in Subsection 2.3. Let X = hX, τ(d)i. Then X is a compact metrizable space having ∞ as its unique accumulation point. Since, in N 2, the set [0, 1]R is well-orderable, while A has no choice function, it follows that X does not embed in the Tychonoff cube [0, 1]R. This shows that the statement “There exists a compact metrizable space with a unique accumulation point which is not embeddable in [0, 1]R” has a permutation model.

6 The list of open problems

For readers’ convenience, let us repeat the open problems mentioned in Sec- tions 4 and 5.

1. Is M(C,WO) equivalent to or weaker than M(TB,WO) in ZF?

2. Does M(C,S) imply CUC in ZF?

3. Does BPI imply CUC in ZF?

4. Does CUC imply M(C,S) in ZF?

R (.(Does M(C, ֒→ [0, 1] ) imply CACfin in ZF? (Cf. Question 5.9(iv .5

References

[1] R. H. Bing, Metrization of topological Spaces, Canadian J. Math. 3 (1951), 175–186.

[2] N. Brunner, Products of compact spaces in the least permutation model, Z. Math. Logik Grundlagen Math. 31 (1985), 441–448.

[3] P. J. Collins and A. W. Roscoe, Criteria for metrisability, Proc. Amer. Math. Soc. 90 (1984), 631–640.

[4] S. M. Corson, The independence of Stone’s theorem from the Boolean prime ideal theorem, https://arxiv.org/pdf/2001.06513.pdf.

41 [5] De la Cruz, E. J. Hall, P. Howard, K. Keremedis and J. E. Rubin, Metric spaces and the axiom of choice, Math. Logic Quart. 49 (2003), 455–466.

[6] De la Cruz, E. J. Hall, P. Howard, K. Keremedis, and J. E. Rubin, Unions and the axiom of choice, Math. Logic Quart. 54 (2008), 652– 665.

[7] E. K. van Douwen, Horrors of topology without AC: A nonnormal orderable space, Proc. Amer. Math. Soc. 95 (1985), 101–105.

[8] R. Engelking, General Topology, Sigma Series in Pure Mathematics 6, Heldermann, Berlin, 1989.

[9] C. Good and I. Tree, Continuing horrors of topology without choice, Topology Appl. 63 (1995), 79–90.

[10] C. Good, I. Tree, W. Watson, On Stone’s theorem and the axiom of choice, Proc. Amer. Math. Soc. 126 (1998), 1211–1218.

[11] E.J. Hall, K. Keremedis and E. Tachtsis, The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters, Math. Logic Quart. 59 (2013), 258–267.

[12] H. Herrlich, Axiom of Choice, Lecture Notes in Mathematics 1875, Springer, New York, 2006.

[13] P. Howard, Limitations on the Fraenkel-Mostowski method of independence proofs, J. Symb. Logic 38 (1973), 416–422.

[14] P. Howard, K. Keremedis, J. E. Rubin and A. Stanley, Paracompact- ness of metric spaces and the axiom of multiple choice, Math. Logic Quart. 46 (2000), 219–232.

[15] P. Howard and J. E. Rubin, Consequences of the axiom of choice, Math. Surveys and Monographs 59, A.M.S., Providence R.I., 1998.

42 [16] P. Howard and J. E. Rubin, Other forms added to the ones from [15], https://cgraph.inters.co/.

[17] P. Howard and J. Solski, The Strength of the ∆-system Lemma, Notre Dame J. Formal Logic 34 (1993), no. 1, 100–106.

[18] P. Howard, D. I. Saveliev, and E. Tachtsis, On the set-theoretic strength of the existence of disjoint coﬁnal sets in posets without max- imal elements, Math. Logic Quart. 62 (2016), no. 3, 155–176.

[19] P. Howard and E. Tachtsis, On metrizability and compactness of cer- tain products without the axiom of choice, submitted.

[20] T. Jech, The Axiom of Choice, North-Holland Publishing Co., 1973.

[21] T. Jech, Set Theory. The Third Millennium Edition, revised and expanded, Springer Monographs in Mathematics, Springer, Berlin, 2003.

[22] K. Keremedis, Consequences of the failure of the axiom of choice in the theory of Lindelöf metric spaces, Math. Logic Quart. 50 (2004), no. 2, 141–151.

[23] K. Keremedis, On sequentially compact and related notions of compactness of metric spaces in ZF, Bull. Pol. Acad. Sci. Math. 64 (2016), 29–46.

[24] K. Keremedis, Some notions of separability of metric spaces in ZF and their relation to compactness, Bull. Pol. Acad. Sci. Math. 64 (2016), 109–136.

[25] K. Keremedis, Clopen ultraﬁlters of ω and the cardinality of the Stone space S(ω) in ZF, Topology Proc. 51 (2018), 1–17.

[26] K. Keremedis and E. Tachtsis, On Loeb and weakly Loeb Hausdorﬀ spaces, Sci. Math. Jpn. Online 4 (2001), 15–19.

43 [27] K. Keremedis and E. Tachtsis, Compact metric spaces and weak forms of the axiom of choice, Math. Logic Quart. 47 (2001), 117–128.

[28] K. Keremedis and E. Tachtsis, Countable sums and products of metrizable spaces in ZF, Math. Logic Quart. 51 (2005), 95–103.

[29] K. Keremedis and E. Tachtsis, Countable compact Hausdorﬀ spaces need not be metrizable in ZF, Proc. Amer. Math. Soc. 135 (2007), 1205-1211.

[30] K. Keremedis and E. Wajch, On Loeb and sequential spaces in ZF, Topology Appl. 280 (2020), 101279.

[31] K. Keremedis and E. Wajch, Cuf products and cuf sums of (quasi)-metrizable spaces in ZF, submitted, preprint available at http://arxiv.org/abs/2004.13097

[32] K. Kunen, Set Theory. An Introduction to Independence Proofs, North-Holland, Amsterdam, 1983.

[33] K. Kunen, The Foundations of Mathematics, Individual Authors and College Publications, London, 2009.

[34] P. A. Loeb, A new proof of the Tychonoﬀ theorem, Amer. Math. Monthly 72 (1965), 711–717.

[35] J. Nagata, On a necessary and suﬃcient condition on metrizability, J. Inst. Polytech., Osaka City University 1 (1950), 93–100.

[36] J. Nagata, Modern General Topology, North-Holland, 1985.

[37] D. Pincus, Zermelo-Fraenkel consistency results by Fraenkel- Mostowski methods J. Symbolic Logic 37 (1972), 721–743.

[38] D. Pincus, Adding Dependent Choice, Annals Math. Logic 11 (1977), 105–145.

44 [39] Y. M. Smirnov, A necessary and suﬃcient condition for metrizability of a topological space, Dokl. Akad. Nauk. SSSR (N.S.) 77 (1951), 197–200.

[40] E. Tachtsis, Disasters in metric topology without choice, Comment. Math. Univ. Carolin. 43 (2002), 165–174.

[41] E. Tachtsis, Inﬁnite Hausdorﬀ spaces may lack cellular families or

inﬁnite discrete spaces of size ℵ0, Topology Appl. 275 (2020), 106997.

[42] S. Willard, General Topology, Addison-Wesley Series in Math., Addison-Wesley Publishing Co., Reading, Massachusetts, 1968.